Analytic Extension of Riemannian Analytic Manifolds and Local Isometries

This article deals with a locally given Riemannian analytic manifold. One of the main tasks is to define its regular analytic extension in order to generalize the notion of completeness. Such extension is studied for metrics whose Lie algebra of all Killing vector fields has no center. The generalization of completeness for an arbitrary metric is given, too. Another task is to analyze the possibility of extending local isometry to isometry of some manifold. It can be done for metrics whose Lie algebra of all Killing vector fields has no center. For such metrics there exists a manifold on which any Killing vector field generates one parameter group of isometries. We prove the following almost necessary condition under which Lie algebra of all Killing vector fields generates a group of isometries on some manifold. Let g be Lie algebra of all Killing vector fields on Riemannian analytic manifold, h ⊂ g is its stationary subalgebra, z ⊂ g is its center and [ g , g ] is commutant. G is Lie group generated by g and is subgroup generated by h ⊂ g . If h ∩ ( z + [ g ; g ] ) = h ∩ [ g ; g ] , then H is closed in G .